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## 5.4 Test on the reference standard deviation

Null hypothesis H_0: m0 = m0’ is tested versus alternative hypothesis H_1: m0 neq m0’. Test criterion is ratio of a posteriori estimate of reference standard deviation

 ``` m0' = sqrt( v'P v / r). ```

and a priori reference standard deviation m0 (input data parameter `m0-apr`). For given significance level alpha lower and upper bounds of interval (L, U) are computed so, that if hypothesis H_0 is true, probabilities P(m0’/m0 le D) and P(m0’/m0 ge H) are equal to alpha/2. Lower and upper bounds of the interval are computed as

 ``` L = sqrt((Chi^2_{1-alpha/2,r})/r), U = sqrt((Chi^2_{ alpha/2 ,r})/r). ```

Probability

 ``` P(L < m0'/m0 < U) = `conf-pr` ```

is by default 95%, this corresponds to 5% confidence level test.

Exceeding the upper limit H of the confidence interval can be caused even by a single gross error (one outlying observation). Method of Least Squares is generally very sensitive to presence of outliers. Safely can be detected only one observation whose elimination leads to maximal decrease of a posteriori estimate of reference standard deviation

 ```(6) m0'' = sqrt{(v'P v - delta)/(r-1)}, delta = max(v_i^2/q_vi), ```

where

 ```(7) q_vi = 1/p_i - q_Li ```

is weight coefficient of i-th residual. If the set of observations contains only one gross error, the outlying observation is likely to be detected, but this can not be guaranteed.

In addition, program `gama-local` computes a posteriori estimate of reference standard deviation separately for horizontal distances and directions and/or angles after formula from

 ``` m0'_t = sqrt(sum{~v^2_it}) / sum{~q_vi}), t=d,s, ```

where symbol t denotes observed distances, directions and/or angles.

## Example

 ```m0 apriori : 10.00 m0' empirical: 9.64 [pvv] : 3.43560e+03 During statistical analysis we work - with empirical standard deviation 9.64 - with confidence level 95 % Ratio m0' empirical / m0 apriori: 0.964 95 % interval (0.773, 1.227) contains value m0'/m0 m0'/m0 (distances): 0.997 m0'/m0 (directions): 0.943 Maximal decrease of m0''/m0 on elimination of one observation: 0.892 Maximal studentized residual 2.48 exceeds critical value 1.95 on significance level 5 % for observation #35 ```

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